Question

Position vs. Time of a Mass-Spring System in Simple Harmonic Motion 0.85. 080 0.75 0.70 13 12 11 10 7 9 6 Times) 5 1. Make a sketch to predict what the graph of position vs. time would look like. There's no need for numbers -just the basic shape, assuming that the motion sens the mass is at its hiahest point.2 pts or clock starts when Pick up the handout of an actual graph of position motion sensor. Make any corrections to the shape of your graph in the previous question without erasing (use a different colored pen). vs. time showing data collected from a By relating the graph to the lab setup on the first page of notes, answer the following questions. Be very accurate by carefully picking numbers from the graph. Show your work in the space provided beneath each question. 2. How far away from the sensor was the mass when at rest (in cm)? +4 pts H= 3. How high did the mass go (in cm)? +2 pts 4. What is the maximum displacement from the resting position, Smax? tApts Smax= 5. Recall the period of the function is the time it takes to complete one cycle (and start repeating). The period T (in seconds) of this motion is: (explain) +2pts T= 6. (a) Would this graph most easily be modeled as a sine function or cosine function? Explain. pts (b) Are there any transformations of the parent trig function? Explain. +2 pfs (c) Using the numbers you found from the previous questions, write a function that accurately models this graph. Explain and justify by graphing your function with a TI84 or with desmos and comparing to the actual graph. Refer to the notes for help with finding the constant b (the number that multiplies the independent variable t). Show the work below and place your final answer in the box. +4 pts Position Function: Calculus Corner Using the recipes you have at your disposal so far, find the derivative of your position function. This will be the velocity function of the mass-spring system. + 2pts Velocity Function: 1. Explain with common sense why "what's missing from s(t)" is missing here. 2pts t 2pfs 2. Make a sketch of this function. Include the numbers from your velocity function.

Answer 1

Answer for calculus part as asked:

(A)

For velocity function:

we can see that the maximum displacement(A) = 0.85-0.75= 0.10

from the image we come to know that difference between 2 successive crests is approx.= 1.2 sec

this is equal to time period(T); i.e. T = 1.2 s

now angular velocity (w)= 2*3.14/T = 6.28/1.2 = 5.23

let displacement(position of mass) = y

since it is SHM, y = A Cos(- wt) = A Cos(wt); (Cosine function used because wave starts at maximum; negative sign because of forward moving wave)

now v =

dy dt

thus v = -A w Sin(wt);

v = -0.1 * 5.23 Sin(5.23t)

thus v = -0.523 Sin (5.23t)

(B)

1. it depends on the context you are asking the question.

like, there is always some damping associated with the shm, which adds a term containing damping constant(b) so that the equation becomes:

y = A Cos(wt) + B e^-bt/2m cos(ωt+φ)

where 1st part is the steady state and second part is transient state.

2.

This is the graph

2 1 AAA 2TT/3 411/3 TT (0.3, -0.523) -1-